Algebra 1 Lesson 1-3. Real Numbers

Transcription

1 Algebra 1 Lesson 1- Common Core Real Numbers and the Number Line Real Numbers Rational Numbers Integers Whole Numbers Irrational Numbers Natural Numbers 1

2 Natural Numbers what you see in nature counting numbers positive numbers any number greater than zero always to the right of zero 0 2

3 Whole Numbers includes all natural numbers zero Integers Includes all whole numbers (0, 1, 2,, ) Negative numbers (opposites of whole numbers) any number less than zero always left of zero

4 Terminating Decimals A number written as a decimal where the number has a limited number of digits. (It has an ending) What are examples? Repeating Decimals A number written as a decimal where there is a repeating pattern of digits that NEVER ends. What are examples? 4

5 Rational Numbers All integers Decimals that have an ending (terminate) OR repeat indefinitely Fractions, a/b where a and b are integers and b 0. Examples:

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7 Irrational Numbers Cannot be written as a fraction (ratio of 2 intergers) Decimal goes on forever without repeating (never ends) Example: Pi You can classify numbers using sets Set: a well-defined collect of terms Example: The set of all integers { -, -2, -1, 0, 1, 2, } Element of the Set: members of a set Example: What are elements of the above set? -1, 1, 100 Subset: elements from a given set Example: What are the natural numbers with the above set of integers? {1, 2, } 7

8 Real Numbers Rational Numbers fractions & terminating or repeating decimals Integers ( -, -2, -1, 0, 1, 2, ) Whole Numbers (0, 1, 2, ) Irrational Numbersdecimals that go on forever that DO NOT repeat Natural Numbers (1, 2, ) I d Rather Be Rational verses Irrational If you could be a number, would you rather be rational or irrational? Write your response on a sticky note, explain why and then post it on the whiteboard. 8

9 Classify each number: Classify each number 9

10 Square Roots An operation that yields a number which, when multiplied by itself produces the given number Example: 16 Radical sign Parts of a Radical 10

11 Inverse Operations (Opposite Operations) What is the opposite of Addition Subtraction Multiplication Squaring a Number Division Square Root Square Roots If a ³0 and a 2 = b, then a is the square root of b. Squares Square Roots 0 2 = 0 0= = 1 1= = 4 4= 2 2 = 9 9= 4 2 = 16 16= 4 What is the relationship between Squares and Square Roots? 11

12 Inverse Operations (Opposite Operations) What is the opposite of Addition Subtraction Multiplication Squaring a Number Division Square Root Square Roots If a ³0 and a 2 = b, then a is the square root of b. Squares Square Roots 0 2 = 0 0= = 1 1= = 4 4= 2 2 Perfect Squaressquare roots that are 9= = = whole 16 numbers 16= 4 (There are 11 of them from 0 to 100. Can you name them?) 12

13 Perfect Square Roots: square If a perfect square roots that have root gives a you whole a number whole answer What happens as an if answer it is NOT 0= 0 1= 1 a perfect square? 25 = 5 It is an irrational 100 = 10 number 6 = 6 4= 2 49 = 7 9= 64 = 8 16 = 4 81 = 9 BrainPop Video & Do It Review Square Roots & Perfect Square Roots perations/squareroots/ 1

14 Try This Use a calculator to approximate the Write the number on a piece of paper Enter it into your calculator, square it, then subtract. Do you get 0? EXPLAIN Watch this video you never know when you may lose out on money because you do not know about square roots! SsQ2I 14

15 Non-Perfect Square Roots Use estimation to figure out the square root of is between what 2 perfect squares? 9 & 16 9 = 16 = 4 Therefore the square root of 14 is a irrational number between & 4. Check your calculator to see the exact irrational number. 15

16 25 & 6 5 & & & & & - - Another way to approximate square roots Watch this video. V_py0ic4ugNlplcXRtdjJtc1k/edit?pli=1 16

17 Try these in your calculator: 25 = = 12 2,104 =

18 Look at these examples again 25 = 5 OR 25 = What is a negative number multiplied by another negative number? -5-5 A positive So how many answers are there to every square root problem? Why? Find the Square Root and then determine if it is rational or irrational

19 Non-Real Numbers What is difference about these examples? Look at these examples again 25 = 5 OR 25 = What is a negative number multiplied by another negative number? -5-5 A positive Therefore, you will always have a positive number under your radical sign. 19

20 Kinds of Square Roots When you first learned Principal Square about Root square roots. You only learned about principle -all #s are positive square roots. Now that you know your Integer Rules Negative Square and Root more about square -negative sign roots you appears know that ALL outside the radical square signroots have 2 answers! 64= 8 64= 8 Both Square Roots -both the negative & positive signs are used = ±8 ± 64 Kinds of Square Roots Principal Square Root: The number that is multiplied by itself is a positive number Example: 64= 8 You typically do not add the positive sign in front of the problem and answer. The positive is implied! 20

21 Kinds of Square Roots Negative Square Root: The number that is multiplied by itself is a negative number (Remember: a negative times a negative is a positive) Example: 64= 8 The negative sign appears outside the radical sign. So your answer must have a negative sign too! Kinds of Square Roots Both Square Roots: The number that is multiplied by itself could be BOTH a negative & positive number Example: ± 64 = ±8 The negative sign and positive signs appear outside the radical sign. So your answer must have BOTH signs! 21

22 Find each square root. If necessary, round to the nearest hundredth ± ± Cube Roots An operation that yields a number which, when multiplied by itself three times produces the given number Example: 8 = 2 What multiplied by itself three times equals 8? Radical sign 22

23 Perfect Cube Roots: cube roots that have a whole number as an answer 0 = = = 10 1 = = 6 8 = 27 = 2 4 = 512 = = = 9 E C A D F B 2

24 What are the symbols of inequality? 24

25 Write >, <, or = to make the sentence true > < =

26 Write each set of numbers in order from least to greatest , 0.6, 2. 8, 4.8, , 0.42, 4 4.8, 0.4, 8 The Code Name Organizer Students will need: Name Decoder Chart Poster Board Markers 26

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